Answer :
We start with the polynomial equation
[tex]$$
x^4 + 10x^3 + 24x^2 - 19x - 70 = 0.
$$[/tex]
Step 1. Factor the polynomial
Notice that the polynomial factors into
[tex]$$
(x + 2)(x + 5)(x^2 + 3x - 7) = 0.
$$[/tex]
Step 2. Set each factor equal to zero
We have three factors. Setting each factor equal to zero gives:
1. [tex]$$x + 2 = 0 \quad \Longrightarrow \quad x = -2.$$[/tex]
2. [tex]$$x + 5 = 0 \quad \Longrightarrow \quad x = -5.$$[/tex]
3. [tex]$$x^2 + 3x - 7 = 0.$$[/tex]
Step 3. Solve the quadratic equation
To solve the quadratic equation
[tex]$$
x^2 + 3x - 7 = 0,
$$[/tex]
we use the quadratic formula:
[tex]$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
$$[/tex]
where [tex]$a=1$[/tex], [tex]$b=3$[/tex], and [tex]$c=-7$[/tex]. This gives:
[tex]$$
x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-7)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 28}}{2} = \frac{-3 \pm \sqrt{37}}{2}.
$$[/tex]
Thus, the quadratic yields two real roots:
[tex]$$
x = \frac{-3 + \sqrt{37}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{37}}{2}.
$$[/tex]
Step 4. Write the complete list of solutions
The real solutions of the original polynomial equation are:
[tex]$$
\boxed{-2,\ -5,\ \frac{-3+\sqrt{37}}{2},\ \frac{-3-\sqrt{37}}{2}}.
$$[/tex]
These are all the real roots of the equation.
[tex]$$
x^4 + 10x^3 + 24x^2 - 19x - 70 = 0.
$$[/tex]
Step 1. Factor the polynomial
Notice that the polynomial factors into
[tex]$$
(x + 2)(x + 5)(x^2 + 3x - 7) = 0.
$$[/tex]
Step 2. Set each factor equal to zero
We have three factors. Setting each factor equal to zero gives:
1. [tex]$$x + 2 = 0 \quad \Longrightarrow \quad x = -2.$$[/tex]
2. [tex]$$x + 5 = 0 \quad \Longrightarrow \quad x = -5.$$[/tex]
3. [tex]$$x^2 + 3x - 7 = 0.$$[/tex]
Step 3. Solve the quadratic equation
To solve the quadratic equation
[tex]$$
x^2 + 3x - 7 = 0,
$$[/tex]
we use the quadratic formula:
[tex]$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
$$[/tex]
where [tex]$a=1$[/tex], [tex]$b=3$[/tex], and [tex]$c=-7$[/tex]. This gives:
[tex]$$
x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-7)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 28}}{2} = \frac{-3 \pm \sqrt{37}}{2}.
$$[/tex]
Thus, the quadratic yields two real roots:
[tex]$$
x = \frac{-3 + \sqrt{37}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{37}}{2}.
$$[/tex]
Step 4. Write the complete list of solutions
The real solutions of the original polynomial equation are:
[tex]$$
\boxed{-2,\ -5,\ \frac{-3+\sqrt{37}}{2},\ \frac{-3-\sqrt{37}}{2}}.
$$[/tex]
These are all the real roots of the equation.
